free vibration of clt plates - adivbois · 2018-07-23 · 2010). floor vibration is taken into...
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17
Rakenteiden Mekaniikka (Journal of Structural Mechanics)
Vol. 47, No 1, 2014, pp. 17-33
Free vibration of CLT plates
Jussi-Pekka Matilainen1 and Jari Puttonen
Summary. This article discusses the ability of different methods to predict the natural frequencies of vibration for a cross laminated timber (CLT) plate. The prediction models considered are the method proposed by Finnish design code for timber structures (RIL), the classical laminated plate theory (CLPT), the first-order shear deformation theory (FSDT) and the finite element method with a shell element (S4R) of Abaqus code. The basic principles governing the dynamic behavior of an orthotropic plate are explained and the effect of shear deformation on the predicted frequencies is illustrated. Constitutive equations for a CLT plate are shown and their implication on the mode shapes is explained. Suitability of the prediction models to the CLT plates are evaluated by comparing them with two different laboratory experiments. First, an experiment carried out by Maldonado & Chui (2012) is illustrated in order to highlight the effect of aspect ratio on natural frequencies. Finally, results of a new experiment conducted for this study demonstrate the effect of side-to-thickness ratio on measured and predicted frequencies.
Key words: cross laminated timber, CLT, vibration, natural frequencies
Received 8 January 2014. Accepted 27 June 2014. Published online 11 July 2014.
Introduction
CLT is a massive timber product, which is used in load bearing applications such as plates
and shear walls. CLT elements are made up of ordinary boards, glued together in a cross-
layered fashion and typically showing symmetric layup (Stürzenbecher, et al., 2010).
Since timber is a relatively soft and lightweight construction material, the design of such
structures is driven by serviceability criteria like maximum deflections and vibration
susceptibility (Gsell, et al., 2007). The serviceability of timber floors becomes more and
more relevant due to the use of high strength materials and longer spans (Hamm, et al.,
2010). Floor vibration is taken into account in the European design code for timber
structures, in which it is required that the fundamental frequency of a timber floor should
be assessed (CEN, 2004). However, the dynamic behavior of CLT floor systems differs
from that of traditional lightweight wood-joisted floors and heavy concrete slabs.
Therefore the existing design codes may not be able to predict the fundamental
frequencies accurately (Hu & Gagnon, 2012). The purpose of this article is to describe
the dynamic behavior of a CLT plate and to assess suitable methods for evaluating the
natural frequencies of a CLT floor.
1Corresponding author. [email protected]
18
Dynamic behavior of a CLT plate
In order to gain a basic understanding of the dynamic behavior of a CLT plate, a brief
review of mechanics of laminated composite plates is required. Reddy (2003) provides a
comprehensive treatment on the subject.
The simplest laminated plate theory is the classical laminated plate theory (CLPT),
which is an extension of the Kirchhoff plate theory to laminated composite plates.
Assumptions of the Kirchhoff plate theory neglect both the transverse shear and normal
stresses. Therefore, the plate deformation is entirely due to bending and in-plane
stretching which implies that the Kirchhoff assumptions are valid only for relatively thin
plates. Practical consequences of the Kirchhoff assumptions lead to overprediction of the
natural frequencies with relatively thick plates because the shear deformations are not
accounted for.
Relaxation of the Kirchhoff assumptions by including a constant state of transverse
shear stresses through the laminate thickness results in the first-order shear deformation
theory (FSDT). FSDT is based on the Mindlin plate theory, which assumes a linear
variation of displacements across the plate thickness while maintaining the transverse
inextensibility of the plate thickness. Since the FSDT takes into account shear
deformation, it is recommended to be used with relatively thick plates. However, the
assumption of constant state of transverse shear stresses violates the boundary conditions
at the plate surfaces where shear stresses should vanish. Therefore, the FSDT requires
shear correction factors to correct the unrealistic variation of the shear stresses through
the thickness. A well-known value for the shear correction factor for homogeneous plates
is 𝐾 = 5 6⁄ , but this value is different for every laminated plate. In order to avoid the use
of shear correction factors, higher-order shear deformation theories (HSDT) have been
developed. They involve higher-order stress resultants that are difficult to interpret
physically and are not dealt with in this article.
From the mechanical point of view, CLT plate is a symmetric cross-ply laminated
plate in which the ply stacking sequence, material properties and geometry are symmetric
about the mid-plane. Because of this symmetry, coupling is eliminated between extension
and shear as well as between bending and twisting. Therefore, the symmetric laminates
have no tendency for twisting, and both the force and moment resultants for a symmetric
laminate have the same form as the orthotropic single-layer plates. The constitutive
equations for a symmetric cross-ply laminate based on the classical and first-order
theories are given by:
{
𝑁𝑥𝑥
𝑁𝑦𝑦
𝑁𝑥𝑦
} = [
𝐴11 𝐴12 0𝐴12 𝐴22 0
0 0 𝐴66
] {
𝜀𝑥𝑥0
𝜀𝑦𝑦0
𝛾𝑥𝑦0
}, (1)
{
𝑀𝑥𝑥
𝑀𝑦𝑦
𝑀𝑥𝑦
} = [
𝐷11 𝐷12 0𝐷12 𝐷22 0
0 0 𝐷66
] {
𝜅𝑥𝑥
𝜅𝑦𝑦
𝜅𝑥𝑦
}, (2)
{𝑄𝑦
𝑄𝑥} = 𝐾 [
𝐴44 00 𝐴55
] {𝛾𝑥𝑦
0
𝛾𝑥𝑧0
}, (3)
19
where the in-plane force resultants are denoted as 𝑁𝑥𝑥, 𝑁𝑦𝑦 and 𝑁𝑥𝑦, the moment
resultants as 𝑀𝑥𝑥, 𝑀𝑦𝑦 and 𝑀𝑥𝑦 and the transverse force resultants as 𝑄𝑥 and 𝑄𝑦. The
mid-plane strains of the laminate are expressed as 𝜀𝑥𝑥0 , 𝜀𝑦𝑦
0 , 𝛾𝑥𝑦0 and 𝛾𝑥𝑧
0 while the
curvatures are expressed as 𝜅𝑥𝑥, 𝜅𝑦𝑦 and 𝜅𝑥𝑦. The extensional and bending stiffnesses are
given by 𝐴𝑖𝑗 and 𝐷𝑖𝑗, respectively, while the shear correction coefficient is given by 𝐾.
From the Equations 1 to 3, it can be seen that the coupling between normal forces and
shearing strain and normal moments and twist is zero (i.e. 𝐴16, 𝐴26, 𝐷16 and 𝐷26 are zero).
In terms of dynamic behavior, this implies that the mode shapes of a CLT plate remain
aligned with the laminate axes. However, in case of generally orthotropic layers (i.e., the
principal material coordinates do not coincide with those of the plate) the mode shapes
tend to twist. The principal material coordinates refer to the directions parallel and
perpendicular to the fibres in an orthotropic layer. Example of this is provided by Figure
1, where the second mode shapes of a specially orthotropic plate and an angle-ply plate
with 45° rotation to the plate coordinate system are compared.
Figure 1. Mode shapes of an angle-ply (left) and specially orthotropic plates (right).
It is of interest to study the dynamic behavior predicted by the CLPT and FSDT of a
specially orthotropic plate through the Navier problem where a rectangular plate is simply
supported on all four edges. The natural frequency of a specially orthotropic single-layer
plate depends on many parameters such as plate aspect ratio (𝑎 𝑏⁄ ), modulus ratio (𝐸1 𝐸2⁄ ) and side-to-thickness ratio (𝑎 ℎ⁄ ). In this context, 𝑎 and 𝑏 denote the in-plane
dimensions along the 𝑥- and 𝑦-coordinate directions of the rectangular laminate while 𝐸1
and 𝐸2 denote the moduli of elasticity in these directions, respectively. The geometry and
coordinate system for a simply supported rectangular plate are shown in Figure 2.
Natural frequencies of an orthotropic plate can be solved as eigenvalues of well-
known free vibration problem
𝑑𝑒𝑡([𝐾] − 𝜔2[𝑀]) = 0, (4)
where [𝐾] is the stiffness matrix, [𝑀] is the mass matrix and 𝜔 is the circular frequency
of vibration.
20
CLPT solution of Equation (4) in terms of cyclic frequencies is given by (Reddy, 2003):
𝑓𝑚,𝑛 =𝜋
2𝑎2√
𝐷11
𝜌ℎ√𝑚4 + [2
𝐷12 + 2𝐷66
𝐷22𝑚2𝑛2 (
𝑎
𝑏)
2
+ (𝑎
𝑏)
4
𝑛4]𝐷22
𝐷11, (5)
where 𝑓 is the cyclic frequency, 𝜌 is the density, ℎ is the plate thickness and 𝑎 and 𝑏
denote the planar dimensions of the plate. The cyclic frequency 𝑓 is related to the circular
frequency by 𝑓 = 𝜔 2𝜋⁄ .
For different values of 𝑚 and 𝑛, there is a unique frequency 𝑓𝑚,𝑛 and a corresponding
mode shape. For plates supported on four sides, 𝑚 and 𝑛 define the number of half sine
waves in the 𝑎 and 𝑏 directions, respectively. The corresponding FSDT solution in terms
of circular frequencies can be found in the work of Reddy (2003) but, for brevity, is not
presented in this article.
In order to assess the applicability of current design practice to the prediction of natural
frequencies of CLT plates, a model used in Finland is taken into consideration. For a
rectangular floor which is simply supported along all four edges, the Finnish Association
of Civil Engineers (RIL, 2009) suggests that the fundamental frequency may
approximately be calculated as
𝑓1 =𝜋
2𝑙2√
(𝐸𝐼)𝑙
𝜌ℎ√1 + [2 (
𝑙
𝐵)
2
+ (𝑙
𝐵)
4
](𝐸𝐼)𝐵
(𝐸𝐼)𝑙, (6)
where the floor spans are given by 𝑙 = 𝑎 and 𝐵 = 𝑏 while the plate bending stiffnesses
of the floor about an axis are given by (𝐸𝐼)𝑙 , (𝐸𝐼)𝐵 = 𝐷11, 𝐷22.
x,
E1
y, E2
b
a
Figure 2. Geometry and coordinate system for a simply
supported rectangular plate.
21
It can be shown that the Equation (6) is derived from the CLPT, Equation (5), with an
assumption that the Poisson’s ratio is assumed constant.
For comparison, the fundamental frequencies are also calculated with
Abaqus/Standard which is a finite element analysis program. The plate is modeled with
general-purpose conventional shell element S4R which allows transverse shear
deformation. S4R stands for a 4-node, quadrilateral, shell element with reduced
integration. (Abaqus, 2012)
For convenience, the following nondimensional fundamental circular frequency is
used in presenting the numerical results in graphical forms in this article:
�̅� = 𝜔𝑏2
𝜋2√
𝜌ℎ
𝐷22, (7)
Figure 3 shows a plot of nondimensionalized fundamental circular frequency �̅� as a
function of plate aspect ratio 𝑎 𝑏⁄ , Figure 4 shows a plot of �̅� as a function of modulus
ratio 𝐸1 𝐸2⁄ and Figure 5 shows a plot of �̅� as a function of side-to-thickness ratio 𝑎 ℎ⁄ .
The following material properties are used to represent typical values for a CLT plate
made of spruce:
𝐸1 𝐸2⁄ = 30, 𝐺12 = 𝐺13 = 1.9𝐸2, 𝜈12 = 0.44, (8)
From Figure 3 it can be seen that all the models used (RIL, CLPT, S4R) behave in the
same manner as function of aspect ratio 𝑎 𝑏⁄ . Clearly, the fundamental frequency
increases as the stiffer side (𝑎) becomes relatively shorter than the softer side (𝑏). Figure
4 demonstrates that the fundamental frequency increases with modular ratio 𝐸1 𝐸2⁄ with all the models used (RIL, CLPT, S4R). Figure 5 reveals the main differences
between the models. Two models (RIL and CLPT) are totally insensitive to side-to-
thickness ratio 𝑎 ℎ⁄ and the only difference between these models is due to the
approximation adopted in the RIL model. Instead, the shear deformation models (S4R
and FSDT) demonstrate that the fundamental frequency begins to decrease with lower
values of 𝑎 ℎ⁄ as the shear deformation steps into the picture.
22
Figure 3. Nondimensionalized fundamental frequency (�̅�) as a function of plate aspect ratio
(𝑎 𝑏⁄ ) for simply supported, orthotropic and single-layer plate.
Figure 4. Nondimensionalized fundamental frequency (�̅�) as a function of modulus ratio
(𝐸1 𝐸2⁄ ) for simply supported, orthotropic and single-layer plate.
0
20
40
60
80
100
0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
𝜔̅
𝒂∕𝒃
RIL
CPT
S4R
0
2
4
6
8
10
1 2 3 4 5 6 7 8 9 10
𝜔̅
𝐸1∕𝐸2
RIL
CPT
S4R
23
Figure 5. Nondimensionalized fundamental frequency (�̅�) versus side-to-thickness ratio (𝑎 ℎ⁄ )
for simply supported, orthotropic and single-layer plate.
Experimental vibration tests
CLT floor tests by Maldonado & Chui (2012)
In order to assess the suitability of the presented methods to predict natural frequencies
of a CLT plate, experimental results are evaluated for comparison. Maldonado & Chui
(2012) performed vibration tests for a 5 layer CLT panels with the following dimensions:
1.020 m x 4.870 m x 0.132 m. A total of four CLT panels were used to construct a 4.870
m x 4.080 m CLT floor in the laboratory. A test where the floor was supported on all four
sides was conducted. The floor width was progressively increased by adding a second,
third and a final fourth panel to the floor specimen. The floor was retested at each floor
width. The adjacent panels were connected by self-tapping screws with 30 cm spacing.
Connection of the adjacent panels is shown in Figure 6. Density of the CLT panel was
measured to 560 kg/m3 while the elastic properties equivalent to a single-layer plate were
evaluated to be 𝐸1=11700 MPa and 𝐸2 = 9000 MPa based on the vibration test. In order
to model the mechanical properties of the panel accurately, the following mechanical
properties are assumed in this article: layers in the longitudinal and transverse directions
are of strength classes C24 and C18 according to RIL (2009), respectively. The assumed
mechanical properties are shown in Table 1. In addition, a shear correction factor of 𝐾 =0.6 was assumed for the laminated plate.
Three different modes of natural vibration (𝑓1,1, 𝑓2,1, 𝑓3,1) were identified with four
different aspect ratios 𝑎 𝑏⁄ . The notation 𝑓𝑚,𝑛 defines the frequency 𝑓 with mode shape (𝑚, 𝑛). For plates supported on four sides, 𝑚 and 𝑛 define the number of half sine waves
in the 𝑎 and 𝑏 directions, respectively. Experimental values for the natural frequencies
from Maldonado & Chui (2012) and results from different prediction methods are shown
4
5
6
7
5 10 15 20 25 30 35 40 45 50
𝜔̅
𝑎∕ℎ
RIL
CPT
S4R
FSDT
24
in Table 2. The corresponding mode shapes are shown in Figure 7. Since the floor width
𝑏 was the only variable in this experimental setup it is interesting to study how the
measured natural frequencies developed as a function of aspect ratio (𝑎 𝑏⁄ ) compared to
the predicted values. The effect of aspect ratio on the nondimensionalized frequencies
�̅�1,1, �̅�2,1 and �̅�3,1 for the experimental as well as predicted values is shown in Figures
8, 9 and 10, respectively.
From the Figures 8, 9 and 10 it is seen that the experimental results behave according
to the theoretical curves shown in Figure 1. This is natural since in this experiment the
natural frequency is noted to decrease as the floor width is increased. In addition, the
shear deformation models FSDT and S4R seem to follow the experimental curve quite
well for every value of 𝑎 𝑏⁄ whereas the models RIL and CLPT seem to overestimate the
natural frequencies. This is attributed to the increase of relative thickness and,
consequently, to the effect of shear deformation. It is also noted that the most distinctive
discrepancy between the experimental and predicted results occurs at the fundamental
frequency �̅�1,1. In this case the floor specimen clearly behaves in a more flexible manner
in comparison with the prediction models. This is assumed to be caused by the connection
configuration shown in Figure 6 which seems to prevent the floor specimen to act in a
completely monolithic way during the first mode of vibration. For higher frequencies the
prediction models become more accurate as the multiple half-sine waves acting upon the
floor enhance interlocking between the panels.
Although the exact material properties of the CLT plate used in the experimental setup
are not known exactly and the material properties used in the models are assumed, the
accuracy of the prediction models is evaluated on the basis of relative difference with the
experimental results. Table 3 shows the relative difference for a certain mode of natural
vibration and the mean error of a prediction model.
From Table 3 it is noted that the predicted fundamental frequency 𝑓1,1 deviates more
from the experimental result for every model. The shear deformation models FSDT and
S4R predict clearly more accurate results overall than the model based on the Kirchhoff
assumption (RIL and CLPT). Considering the initial uncertainty in the material
parameters, the overall accuracy of the shear deformation models with approximately 10
% difference with the experimental results seems acceptable. On the other hand, the code
based model (RIL) produces an error of about 30 % and therefore it is questionable if the
model is suitable to predict the natural frequencies of a CLT floor.
25
Table 1. Mechanical properties of the CLT panel with strength classes C24 and C18 (RIL,
2009).
Layer Thickness
(mm)
𝐸1
(MPa)
𝐸2
(MPa)
𝜈12
𝐺12
(MPa)
𝐺13
(MPa)
𝐺23
(MPa)
1 26.4 11000 370 0.44 690 690 50
2 26.4 300 9000 0.015 560 50 560
3 26.4 11000 370 0.44 690 690 50
4 26.4 300 9000 0.015 560 50 560
5 26.4 11000 370 0.44 690 690 50
Figure 6. CLT panel connection configuration during the experiment. (Maldonado &
Chui, 2012)
Figure 7. Mode shapes of frequencies 𝑓1,1, 𝑓2,1 and 𝑓3,1 of CLT floor (four panels) supported
on four sides.
26
Table 2. Experimental values from Maldonado & Chui (2012) and results from different
prediction method
𝑎
𝑏
f1,1
(Hz)
f2,1
(Hz)
f3,1
(Hz)
Results
4.8 79 102.75 126 Mald
onad
o&
Chui (2
012)
2.4 22 47.25 79.25
1.6 15.75 42 76.25
1.2 12 39.5 74
4.8 118.8 137.9 176.6 RIL
205
-1-
20
09 2.4 34.5 59.5 107.2
1.6 19.6 47.6 97.1
1.2 14.9 44.0 93.9
4.8 117.3 132.5 167.2 CL
PT
2.4 33.1 56.4 103.3
1.6 18.6 45.9 95.2
1.2 14.1 43.0 92.8
4.8 83.3 97.3 129.2 FS
DT
2.4 29.9 52.2 94.8
1.6 17.8 44.1 89.0
1.2 13.8 41.6 87.2
4.8 91.9 101.1 121.4 S4
R
2.4 30.7 49.3 82.3
1.6 17.5 40.7 76.3
1.2 13.4 38.3 74.6
27
Figure 8. The effect of aspect ratio on nondimensionalized frequencies �̅�1,1.
Figure 9. The effect of aspect ratio on nondimensionalized frequencies �̅�2,1.
0,0
0,5
1,0
1,5
2,0
2,5
1,0 2,0 3,0 4,0 5,0
𝜔̅ 1,
1
𝒂∕𝒃
EXP.
RIL
CLPT
S4R
FSDT
0,0
2,0
4,0
6,0
8,0
1,0 2,0 3,0 4,0 5,0
𝜔̅ 2,
1
𝒂∕𝒃
EXP.
RIL
CLPT
S4R
FSDT
28
Figure 10. The effect of aspect ratio on nondimensionalized frequencies �̅�3,1.
Table 3. Relative difference in comparison to the mode 𝑓𝑚,𝑛 (left) and mean difference of the
prediction model (right).
Mode
fm,n
RIL
(%)
CLPT
(%)
FSDT
(%)
S4R
(%)
Model Difference
(%)
f1,1 38.9 33.6 17.3 19.7 RIL 30.9
f2,1 21.3 16.6 6.5 7.4 CLPT 26.2
f3,1 32.4 28.3 14.2 2.1 FSDT 12.7
S4R 9.7
0,0
5,0
10,0
15,0
1,0 2,0 3,0 4,0 5,0
𝜔̅ 3,
1
𝒂∕𝒃
EXP.
RIL
CLPT
S4R
FSDT
29
New tests for the CLT plates
Because the exact material properties were not presented in the test carried out by
Maldonado & Chui (2012), additional vibration tests with known material properties were
conducted in this study. In these experiments, two different rectangular CLT plates were
supported on all four sides and the free vibration phenomenon was achieved by impacting
the plates with a hammer. Vertical acceleration history of the plates was recorded at three
different locations in order to extract modes of vibration in the longitudinal direction.
Both the CLT plates had the same in-plane dimensions of 2.45 m x 4.00 m, a density of
470 kg/m3 and were made of strenght class C24 according to RIL (2009). The first plate,
denoted here as CLT 100, was 100 mm thick and consisted of three layers with
thicknesses of 30/40/30 mm and a lamination scheme of 90°/0°/900. The lamination
scheme represents the orientation of a layer to the longer direction of a plate. The second
plate, denoted here as CLT 200, was 200 mm thick and consisted of seven layers with
thicknesses of (20/40/20/40)𝑆𝑂 mm and a lamination scheme of (0°/90°/00/90°)𝑆𝑂.
In-plane dimensions of the plates and illustrations of the accelerometers are shown in
Figure 11.
The vertical acceleration histories were converted into frequency domain by using the
fast Fourier transform (FFT) algorithm in order to identify the natural frequencies. For
both the CLT 100 and CLT 200 plates, three lowest frequencies, namely 𝑓1,1, 𝑓2,1 and 𝑓3,1,
were measured and then compared to the values predicted by the models. The Fourier
spectra for the CLT 100 and CLT 200 plates showing the three lowest natural frequencies
are shown in Figures 12 and 13, respectively. The measured frequencies and results from
the different prediction models as well as the mean errors for the CLT 100 and CLT 200
plates are shown in Tables 4 and 5.
From Figures 12 and 13, the measured frequencies are easily distinguished as peaks
in the frequency domain. Similar plots were generated from numerous vibration events
and therefore the peaks are considered to genuinely represent the natural frequencies of
the plates. It is also noted that for the thinner plate CLT 100 the fundamental frequency
is higher despite the fact that it is only half as thick as the CLT 200. This is because its
top and bottom layers are parallel to the shorter side of the plate and underlines the
importance of adding stiffness in the shorter direction of the plates supported on four
sides.
From Table 4 it is noted that all the prediction models produce extremely accurate
results (error ~ 5 %) for the CLT 100 plate in comparison with the experimental results.
The accuracy of the results is attributed to the relative slenderness (b/h = 25) of the plate
which implies that the shear deformation is not significant.
As can be seen from Table 5, the accuracy of the prediction models drops dramatically
(error ~ 15-70 %) for the CLT 200 plate even though the lower characteristic values were
used for the elastic constants. This is because the side-to-thickness ratio (b/h=12)
decreases and the shear deformation becomes significant. On the other hand, it is possible
that for the multilayer plate the layers are not perfectly bonded together and result in a
lower stiffness than expected. The models that take into account the shear deformation
(FSDT and S4R) perform clearly better than the other models (RIL and CLPT). Again, it
is noted that the model recommended by the design code (RIL, 2009) should be used with
caution with relatively thick CLT plates.
30
Figure 12. Fourier spectrum for the CLT 100 plate.
0
0,1
0,2
0,3
0,4
10 20 30 40 50 60 70
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒
𝑓 [Hz]
CLT 100
34 Hz
y
x
Lenght = 4.000m
Wid
th =
2.4
50 m
Accelerometers
44 Hz
63 Hz
Figure 11. In-plane dimensions and illustration of the accelerometers.
31
Figure 13. Fourier spectrum for the CLT 200 plate.
Table 4. Experimental values and results from different prediction methods for the CLT 100.
Mode
fm,n
RIL
(Hz)
CLPT
(Hz)
FSDT
(Hz)
S4R
(Hz)
Exp.
(Hz)
Model Difference
(%)
f1,1 37.1 37.7 36.7 34.1 34.0 RIL 4.9
f2,1 44.0 46.0 44.8 41.5 44.0 CLPT 4.5
f3,1 59.9 63.2 61.4 58 63 FSDT 3.4
S4R 5.5
Table 5. Experimental values and results from different prediction methods for the CLT 200.
Mode
fm,n
RIL
(Hz)
CLPT
(Hz)
FSDT
(Hz)
S4R
(Hz)
Exp.
(Hz)
Model Difference
(%)
f1,1 57.8 49.2 46.5 43.4 32.0 RIL 71.8
f2,1 107.8 89 79.2 70.9 71.0 CLPT 46.8
f3,1 192.3 169.3 134.2 114.1 105.0 FSDT 28.2
S4R 15.2
0
0,05
0,1
0,15
10 20 30 40 50 60 70 80 90 100 110 120
𝐹𝑜𝑢𝑟𝑖𝑒𝑟
𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒
𝑓 [Hz]
CLT 200
32 Hz 71 Hz
105 Hz
32
Conclusion
The basic principles concerning the dynamical behavior of a simply supported specially
orthotropic plate were explained through various prediction models. Mode shapes of a
CLT floor remain aligned with the laminate coordinate system because the stiffness
matrix coefficients related to twisting moments disappear with the symmetric laminates
with multiple specially orthotropic layers. Natural frequencies increase as the plate aspect
ratio (𝑎 𝑏⁄ ) decreases and the modulus ratio (𝐸1 𝐸2⁄ ) increases. The relative thickness (𝑎 ℎ⁄ ) has an important effect on the natural frequencies even at moderate values (𝑎 ℎ⁄ = 20). Negligence of this leads to overprediction of natural frequencies and in the
worst case to serviceability problems with CLT timber floors.
The prediction models were evaluated by comparing them with two laboratory
experiments. Since the exact material properties were not known in the first experiment,
they were assumed. In the second experiment, two CLT plates with known material
properties were tested. Two models (RIL and CLPT) produced results with a notable
disagreement with the test results in cases where the shear deformation was significant
while the other two models which take into account the shear deformation (FSDT and
S4R) agreed quite well with the results. The code based method (RIL) should be used
with awareness of its range of applicability for relatively thick CLT floors which tend to
be prone to remarkable shear deformation. The shear deformation model FSDT could be
a suitable model for vibration analysis for CLT plates but the effect of shear correction
factors should be studied in more detail. Finite element model with a shell element which
allows shear deformation (S4R) performed in a proper way in every aspect.
In conclusion it can be said that the vibration problem of a CLT plate is a difficult task
to formulate in a design code friendly format because of its compliancy to shear
deformation. It might also be important to study whether the multilayer CLT plates
actually behave in a fully composite manner. However, as illustrated in this study, more
refined plate theories or finite element analysis programs can be used to predict the
dynamical behavior of CLT structures accurately.
Acknowledgements
This study was funded by Aalto University, School of Engineering, Department of Civil
and Structural Engineering. Chief Engineer Veli-Antti Hakala and Laboratory Manager
Jukka Piironen were of great importance during the arrangement of laboratory
experiments.
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Jussi-Pekka Matilainen, Jari Puttonen
Aalto University, School of Engineering,
Department of Civil and Structural Engineering
PO Box 12100
FI - 00076 Aalto